Answer:
C
Step-by-step explanation:
Use Δ ABC to find AB then Δ ABD to find DB
Δ ABC is isosceles since ∠ CAB = ∠ CBA = 45° , then CB = CA = 2
Using Pythagoras' identity in Δ ABC , then
AB² = 2² + 2² = 4 + 4 = 8 ( take square root of both sides )
AB = [tex]\sqrt{8}[/tex] = 2[tex]\sqrt{2}[/tex]
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Using the sine ratio in Δ ABD with the exact value
sin60° = [tex]\frac{\sqrt{3} }{2}[/tex] , then
sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{AB}{DB}[/tex] = [tex]\frac{2\sqrt{2} }{DB}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )
[tex]\sqrt{3}[/tex] × DB = 4[tex]\sqrt{2}[/tex] ( divide both sides by [tex]\sqrt{3}[/tex] )
DB = [tex]\frac{4\sqrt{2} }{\sqrt{3} }[/tex] ( multiply numerator/ denominator by [tex]\sqrt{3}[/tex] to rationalise denominator )
= [tex]\frac{4\sqrt{2} }{\sqrt{3} }[/tex] × [tex]\frac{\sqrt{3} }{\sqrt{3} }[/tex]
= [tex]\frac{4\sqrt{6} }{3}[/tex] → C
